The genus and the category of configuration spaces

TL;DR

This paper investigates the topological and algebraic properties of configuration spaces of smooth manifolds, focusing on group actions and deriving lower bounds for various invariants to understand their structure better.

Contribution

It provides new estimates for the cohomological index, genus, and equivariant LS-category of configuration spaces under group actions, especially p-tori.

Findings

Lower bounds for cohomological index and genus are established.

Estimates for equivariant LS-category are derived.

Corollaries for functions on configuration spaces are obtained.

Abstract

In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly $p$-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel'skii-Schwarz, and the equivariant Lyusternik-Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced.